Hereditary integrals

Memory Effects Hereditary Integrals and Linear Response Theory... 

Notice that the factor is a constant in the first integral, which allowed us to write it as part of the integrand. The second integral is obtained from the first by making the substitution s = t — tf si often referred to as the time lapse). As seen from equation (15), the response of the system is completely characterized by the function i/r(t) = (C/t)g /, which is therefore called the response function (or sometimes the aftereffect function to highlight the fact that the response is delayed). Moreover, it is seen that the (delayed) response may be expressed as a convolution or hereditary integral. 

Many processes in pharmaceutics are related to transport, and the appHcations of the outlined theory are therefore numerous. Notwithstanding their practical importance, the special instances of the general transport equation (11) listed in Table 2 are assumed to be relatively familiar, and will therefore not be discussed further in this chapter. Instead, we focus our attention on applications of hereditary integrals and linear response theory, in particular on dynamic mechanical analysis (DMA) and impedance spectroscopy. 

Equation (10) cannot be applied until A, the equivalent relaxation time for the fluid, is known. However, A is defined by the linear Maxwell model, and actual polymer solutions exhibit marked nonlinear viscoelastic properties [5,6,7]. For both fresh and shear degraded solutions of Separan AP 30 polyacrylamide, which exhibit pronounced drag reduction in turbulent flow, Chang and Darby [8] have measured the nonlinear viscosity and first normal stress functions, and Tsai and Darby [6] have reported transient elastic properties of similar solutions, A nonlinear hereditary integral function containing six parameters has been proposed to represent the measured properties [8], The apparent viscosity function predicted by this model is ... 

The general approach to discussing linear viscoelasticity comes from the Boltzmann superposition principle represented as a hereditary integral. For the shear stress as a function of shear strain, one obtains... 

And integration of the hereditary integrals for strain and deflection gives the solution to any applied history of the moment M t). A note of caution, however, arises for mixed conditions in which the interface between the stress and the displacement boundaries is not constant. In such cases the elastic-viscoelastic correspondence principle is not applicable and the solutions become more difficult (21). 

The viscoelastic response is commonly described by using a form of Boltzmann "s hereditary integral, referred to as the excitation-response theory (Gurtin and Sternberg 1962 Yamamoto 1972 Christensen 2003). 

Finally, we can see the relationship between the response given by the hereditary integral form and that given by conventional creep laws such as the logarithmic form... 

The nonlinear constitutive law due to Schapery may be linearized by assuming that the nonlinearizing parameters 8 y d g2 have a value of unity. In addition, the stress-dependent part of the exponent in the definition of the shift function is set to zero. Consequently, the constitutive law reduces to the hereditary integral form commonly used to describe a linear viscoelastic material. 

In applied viscoelasticity not all the constitutive equations are formulated by an a-priori defined internal energy y/, but the constitutive model is expressed directly by the functional relation between the stress and the strain through an hereditary integral. In rheology this class of constitutive models is called Rivlin-Sawyers models Fxmg s [164], Fosdick and Yu s [165] and many other models currently used belong to this constitutive class. 

Single hereditary formulation has proven to reproduce all the crucial aspects of rubber behavior (hysteresis, relaxation and creep). In the simplest situatirm, the current value of the stress is the sum of two different contributions a purely elastic term depending oti the current value of the strain and a hereditary integral depending oti the whole strain history. In the linear model of viscoelasticity, introduced by Bernstein et al. [166], the stress dependence on the strain histoiy is assumed to be linear, i.e.. 

As discussed previously, the relation between stress and strain for linear viscoelastic materials involves time and higher derivatives of both stress and strain. While the differential equation method can be quite general, a hereditary integral method has proved to be appealing in many situations. This hereditary integral equation approach is attributed to Boltzman and was only one of his many accomplishments. In the late nineteenth century, when the method was first introduced, considerable controversy arose over the procedure. Now, it is the method of choice for the mathematical expression of viscoelastic constitutive (stress-strain) equations. For an excellent discussion of these efforts of Boltzman, see Markovitz (1977). 

Because all events over the history of a viscoelastic material contribute to the current state of stress and strain, the lower limit of the hereditary integral is most often taken to be - < and Elqs. 6.8 and 6.12 therefore become. 

As many nonlinear approaches are beyond the intended level and scope of this text, the focus will be on single integral mathematical models which are an outgrowth of linear viscoelastic hereditary integrals and lead to an extended superposition principle that can be used to evaluate nonlinear polymers. The emphasis will be on one-dimensional methods but these can be readily extended to three dimensions using deviatoric and dilatational stresses and strains as was the case for linear viscoelastic stress analysis as discussed in Chapters 2 and 9. 

Single Integral Approaches Leadermann (1948) recognized the nonlinear nature of polymers and suggested an approach based on a linear hereditary integral given by,... 

Integrals over the history of strain (or stress) as occur in (1.2.1,2) are sometimes referred to as hereditary integrals. Materials whose constitutive equations contain such hereditary integrals are described as having memory. 

If the hereditary integral is absent in (1.2.4), the relation reduces to Hooke s Law where the modulus is time-dependent. Note that the dimension of G t, V) is the same as the moduli in elastic theory, namely force per unit area. 

Note that the Correspondence Principle does not apply if the material is aging. This includes the case where temperature variation destroys the convolution form of the hereditary integrals. However, an extension of the proportionality assumption, described in Sects. 1.8, 1.9, provides for aging materials with only one independent relaxation function. For such materials there exist useful analogies between viscoelastic and appropriate elastic solutions. These are referred to in Sect. 2.12. 

It is on these, rather than (2.3.8, 12), that the approach developed in the following sections will rest on these equations and (2.3.9) or (2.3.13). We remark that relations (2.3.9) and (2.3.13) could have been written down directly, at least if the proportionality assumption is made. Essentially, the point is that made in the context of (1.8.23) which we restate here within the present simpler, more concrete framework. Consider (2.3.9) for example. The proportionality assumption means that there is only one hereditary integral in the theory, and the equations of the theory are identical to the elastic equations if displacements are replaced by quantities of the form of v(r, t) where l t) is proportional to //(/), for example. It follows that elastic solutions are applicable if displacements are replaced by i (r, t) and corresponding quantities for the other components. This is precisely the content of equation (2.3.9). A similar argument applies to (2.3.13). It is not necessary even to assume proportionality for certain special problems, though these problems are difficult to characterize in fundamental terms. They are mainly problems where all the dependence on material properties can be grouped into one function. 

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